On polynomial representations of Boolean functions related to some number theoretic problem

i1 ,...,in in 1 Ai1 ,...,in xi 1 . . . xn ,
Ai1 ,...,in = 0.
For a real w we define the sign-function as sign w = 1, 0, if w ≥ 0, if w < 0.
Here we also obtain lower bounds on the degree deg f and sparsity spr f of polynomials f providing a threshold representation of S (x) for n -bit integers x , that is a representation of the form sign f (x1 , . . . , xn ) = S (x), where x = x1 . . . xn is the bit representation of x , 1 ≤ x ≤ 2n − 1 .
where x = x1 . . . xn is the bit representation of x . For Boolean inputs we simply need to consider multilinear polynomials. Each polynomial over Z Z M is of the form P (x1 , . . . t by NSF grant CCR-9877150. Supported in part by ARC grant A69700294.
For a given integer n ≥ 1 , we can identify x , 0 ≤ x ≤ 2n − 1 , and its bit representation x1 . . . xn (if necessary we add several leading zeroes) and consider S (x) as a Boolean function of n variables. We say a polynomial P over Z Z M strongly M -represents S if for all 1 ≤ x ≤ 2n − 1 , P (x1 , . . . , xn ) ≡ S (x) (mod M ), (1) where x = x1 . . . xn is the bit representation of x . Similarly, we say a polynomial P over Z Z M one-sidedly M -represents S if for all 1 ≤ x ≤ 2n − 1 , P (x1 , . . . , xn ) ≡ 0 (mod M ) ⇐⇒ S (x) = 0, (2)
1
2
Abstract. We say a polynomial P over Z Z M strongly M -represents a Boolean function F if F (x) ≡ P (x) (mod M ) for all x ∈ {0, 1}n . Similarly, P one-sidedly M -represents F if F (x) = 0 ⇐⇒ P (x) ≡ 0 (mod M ) for all x ∈ {0, 1}n . Lower bounds are obtained on the degree and the number of monomials of polynomials over Z Z M , which strongly or one-sidedly M -represent the Boolean function deciding if a given n bit integer is square-free. Similar lower bounds are also obtained for polynomials over the reals which provide a threshold representation of the above Boolean function.
1
Introduction
In this paper, we obtain lower bounds on the degree and the number of monomials of polynomials over Z Z M , which strongly or one-sidedly M -represent the Boolean function deciding if a given n -bit integer is square-free. These results provide the first non-trivial lower bounds over Z Z M on the complexity of a number theoretic problem which is closely related to the integer factorization problem. Similar lower bounds are also obtained for polynomials over the reals which provide a threshold representation of the above Boolean function. We also show that some simple number theoretic observations allow us to obtain quite strong lower bounds on several other complexity characteristics of testing if a given integer is square-free. We recall that an integer x is called square-free if there is no prime p such that p2 |x . Otherwise, x is called square-full . We define the function S (x) = 1, 0, if x is square-free, if x is square-full.
H ∈H
AH
i∈H
xi ,
(3)
where H ⊆ 2{1,2,...,n} and 0 = AH ∈ Z ZM. We call the largest value of |H | in the representation (3) the degree of P and write deg P . We call the number of coefficients AH , or equivalently |H| , the sparsity of P and write spr P . In this paper, we obtain lower bounds on the degree deg P and the sparsity spr P of polynomials over Z Z M , satisfying either (1) or (2) for all inputs. Similarly to the case of polynomials over Z Z M , for a polynomial f in n variables over the reals IR, we define the total degree deg f as the largest sum i1 + . . . + in and the sparsity spr f as the number of coefficients Ai1 ...in in the representation f (x1 , . . . , xn ) =
Furthermore, in the case of real polynomials, the Boolean values 0 and 1 can be interpreted as two arbitrary real values α0 and α1 , not necessarily α0 = 0 and α1 = 1 . It is easy to see that the degree of the corresponding polynomials does not depend on the particular choice of α0 , α1 because they are equivalent under a linear transformation of variables [19]. But it is shown in [19] that the sparsity spr f depends on the choice of α0 and α1 . In fact, there are examples of Boolean functions demonstrating that for (α0 , α1 ) = (0, 1) and (α0 , α1 ) = (1, −1) the gap between the numbers of monomials of the corresponding polynomials for these two representations can be exponentially large [19]. Threshold representations of Boolean functions via real polynomials have been studied in a number of works [8, 9, 14, 19, 24, 28]. These papers contain many general estimates together with lower bounds for some particular Boolean functions. However, these Boolean functions are usually specially constructed examples which are not related to any particular number theoretic or combinatorial problem. Representations of Boolean functions via polynomials over Z Z M have been studied in [2, 3, 15, 30]. In these papers, lower and upper bounds are obtained for polynomials representing the OR, MOD M (that determines if the sum of the inputs is not divisible by M ), and ¬ MOD M Boolean functions. We note that a polynomial of degree d over Z Z M is represented by a circuit consisting of an unbounded fan-in MOD M gate at the top where each input wire is a function of no more than d variables. In [12, 29], some lower bounds are obtained for polynomials over Z Z 2 strongly 2 -representing the Boolean function deciding the quadratic residuacity of an n -bit integer x . In the series of papers [4–7] lower bounds have been obtained on the circuit complexity, sensitivity, degree of polynomial representation and other complexity characteristics of testing square-free numbers and computing the greatest common divisor. As in [12, 29] the method of [4–7] is based on the uniformity of distribution of long patterns of 0, 1 in the values of S (x). For the quadratic residuacity a similar property has been established in [12, 29] by using the very powerful Weil estimate, in [4–7] a sieve method has been used for this purpose. In particular, for a strongly 2 -representing polynomial P the lower bound deg P ≥ 0.165 . . . n has been obtained in [5]. It has also been applied to obtain a lower bound of order n1/2 on the degree of real polynomials P which approximate S in the following sense: for all 1 ≤ x ≤ 2n − 1 , |S (x) − P (x1 , . . . , xn )| ≤ 1/3 where x = x1 . . . xn is the bit representation of x . These lower bounds are derived from the asymptotic formula for the sensitivity of the function S obtained in [5]. Unfortunately, there is no link between the sensitivity and the degrees of M -representing polynomials, M ≥ 3 , and of threshold representations. Alternative methods of [1] and [32] yield stronger but less explicit complexity results (which apply to primality testing as well). However these approaches work neither for M -representing polynomials nor for threshold representations.